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2 IMPLICIT FUNCTIONS 273

Rⁿ (since all derivatives of g at (X_Q , j₀) are equal to the corresponding deriva-
tives of/); hence, by (9.3.5.1), v maps a neighborhood of X_Q in R^m into the
space Rⁿ, and the uniqueness part of (10.2.1) therefore proves that the
restriction of v to W n R^m is identical to u. Q.E.D.

One of the most important applications of (10.2.1) is the following:

(10.2.5) Let E, F be two Banach spaces, f a continuously differentiable
mapping of a neighborhood V of x₀ e E into F. If'f^f(x₀) is a linear homeo-
morphism ofE onto F, there exists an open neighborhood U c V ofx₀ such that
the restriction off to U is a homeomorphism of U onto an open neighborhood
of y₀ =^zf(xo) in F. Furthermore, iffisp times continuously differentiable in
U (resp. analytic in U, E and F being finite dimensional), the inverse mapping
9 0//(U) onto U is p times continuously differentiable (resp. analytic) /«/(U).

Apply (10.2.1) to the function h(x, y) =/(x) — y, exchanging the roles of
x and y; as D^XQ , y₀) =/'(^o)? ^we conclude that there is an open ball W
of center y₀ in F and a continuous mapping # of W into E such that^(W) c U,
f(g(y)) = y in W and #(y₀) = x₀; furthermore, by (10.2.3) (resp. (10.2.4)), if
/is/? times continuously differentiable (resp. analytic),#is/? times continuously
differentiable (resp. analytic). From the identity f(g(y)) = y it follows that g
is injective in W, hence is a bijective continuous mapping of W onto
V = #(W) c U; moreover, #(W) =/~¹(W) is open in E, and /is a homeo-
morphism of V = g(W) onto W, which ends the proof.

PROBLEMS

1. Let E, F be two Banach spaces, A an open neighborhood of a point JC_Q e E,/a con-
tinuous mapping of A into F, which is differentiable at X_Q (but not necessarily at other
points of A). Suppose /'(*(>) is ^a linear homeomorphism of E onto its image in F;
show that there is a neighborhood U <= A of x₀ such that f(x) ^ f(x₀) for every
xe\J such that jc ^ JC_Q . (Observe that the assumption implies the existence of a
constant c> 0 such that \\f'(x_Q) - s\\ ^ c\\s\\ for all s e E (5.5.1).)

2. Let /= (/i,/₂) be the mapping of R² into itself defined by fi(x^, x₂) = xi; f₂(xi, x₂) =
x₂ ~ x\ for x\ ^ x₂, jf₂(xi, x₂)« (x₂ - x\x₂)lxl for 0^x₂^ x\, and finally
/2(*i, — *a) = —fz(x\,X2) for x₂ ^ 0. Show that / is differentiable at every point of
R²; at the point (0, 0), D/ is the identity mapping of R² onto itself, but D/ is not
continuous. Show that in every neighborhood of (0,0), there are pairs of distinct points
*', x" such that/(xO = /(^0 (compare to (10.2.5)).



	2 IMPLICIT FUNCTIONS 273 Rⁿ (since all derivatives of g at (X_Q , j₀) are equal to the corresponding deriva- tives of/); hence, by (9.3.5.1), v maps a neighborhood of X_Q in R^m into the space Rⁿ, and the uniqueness part of (10.2.1) therefore proves that the restriction of v to W n R^m is identical to u. Q.E.D. One of the most important applications of (10.2.1) is the following:

	(10.2.5) Let E, F be two Banach spaces, f a continuously differentiable mapping of a neighborhood V of x₀ e E into F. If'f^f(x₀) is a linear homeo- morphism ofE onto F, there exists an open neighborhood U c V ofx₀ such that the restriction off to U is a homeomorphism of U onto an open neighborhood of y₀ =^zf(xo) in F. Furthermore, iffisp times continuously differentiable in U (resp. analytic in U, E and F being finite dimensional), the inverse mapping 9 0//(U) onto U is p times continuously differentiable (resp. analytic) /«/(U). Apply (10.2.1) to the function h(x, y) =/(x) — y, exchanging the roles of x and y; as D^XQ , y₀) =/'(^o)? ^we conclude that there is an open ball W of center y₀ in F and a continuous mapping # of W into E such that^(W) c U, f(g(y)) = y in W and #(y₀) = x₀; furthermore, by (10.2.3) (resp. (10.2.4)), if /is/? times continuously differentiable (resp. analytic),#is/? times continuously differentiable (resp. analytic). From the identity f(g(y)) = y it follows that g is injective in W, hence is a bijective continuous mapping of W onto V = #(W) c U; moreover, #(W) =/~¹(W) is open in E, and /is a homeo- morphism of V = g(W) onto W, which ends the proof.

	PROBLEMS

	1. Let E, F be two Banach spaces, A an open neighborhood of a point JC_Q e E,/a con- tinuous mapping of A into F, which is differentiable at X_Q (but not necessarily at other points of A). Suppose /'((>) is ^a linear homeomorphism of E onto its image in F; show that there is a neighborhood U <= A of x₀ such that f(x) ^ f(x₀)* for every xe\J such that jc ^ JC_Q . (Observe that the assumption implies the existence of a constant c> 0 such that \\f'(x_Q) - s\\ ^ c\\s\\ for all s e E (5.5.1).) 2. Let /= (/i,/₂) be the mapping of R² into itself defined by fi(x^, x₂) = xi; f₂(xi, x₂) = x₂ ~ x\ for x\ ^ x₂, jf₂(xi, x₂)« (x₂ - x\x₂)lxl for 0^x₂^ x\, and finally /2(i, — a) = —fz(x\,X2) for x₂ ^ 0. Show that / is differentiable at every point of R²; at the point (0, 0), D/ is the identity mapping of R² onto itself, but D/ is not continuous. Show that in every neighborhood of (0,0), there are pairs of distinct points *', x" such that/(xO = /(^0 (compare to (10.2.5)).